Let us finally turn to the third claim. As in the proof of Theorem REF  (see e.g. [1]}), we start with writing
\(\phi _{n}(t) = 1 + {i\hspace{0.55542pt}}\mathbb {E}[\xi ^{(b_n)}] t - \mathbb {E}[(\xi ^{(b_n)})^{2}] \frac{t^{2}}{2} + R^{n}_{1}(t)\qquad \text{with}\qquad |R^{n}_{1}(t)| \leqslant \frac{|t|^{3}}{6} \mathbb {E}[|\xi ^{(b_n)}|^{3}].\) 
